---
tag: 43J9XSE6M8
kind: definition
name: Barnes integral
msc: [33C60, 33C05, "30E20"]
defines: 1PDGBGEHM2
uses: [0H5SYBSED9, 06ZGPYTWK8, 463X5V31D9]
sources:
  - url: https://en.wikipedia.org/wiki/Barnes_integral
---

## Definition

Let $a_1, \dots, a_m$, $b_1, \dots, b_n$, $c_1, \dots, c_p$ and $d_1, \dots, d_q$ be complex numbers, and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Assume that $a_j + b_k$ is not in $\set{0, -1, -2, \dots}$ for all $j$ and $k$.
Then the poles $s = -a_j - \ell$ of the factors $\Gamma(a_j + s)$ are disjoint from the poles $s = b_k + \ell$ of the factors $\Gamma(b_k - s)$, where $\ell \in \N$.
Let $L$ be a path from $-i\infty$ to $+i\infty$.
Assume that all poles of the first family lie on its left and all poles of the second family on its right.
Fix $z \ne 0$ and a branch of $\log z$, and put $z^{-s} = e^{-s \log z}$.
A *Barnes integral* is a [[contour integral|06ZGPYTWK8]] of the form
$$\frac{1}{2\pi i} \int_L \frac{\prod_{j=1}^{m} \Gamma(a_j + s) \prod_{k=1}^{n} \Gamma(b_k - s)}{\prod_{j=1}^{p} \Gamma(c_j + s) \prod_{k=1}^{q} \Gamma(d_k - s)}\, z^{-s}\, ds,$$
for those $z$ for which it converges.

## Remarks

Example: take $m = 1$, $a_1 = 0$, no other factors, $x > 0$, and $L$ the line $\operatorname{Re} s = c$ with $c > 0$.
All poles $s = 0, -1, -2, \dots$ of $\Gamma(s)$ lie to the left of $L$, so $L$ is admissible.
Shifting the line to the left and summing the residues $(-1)^\ell x^\ell / \ell!$ at $s = -\ell$ gives
$$\frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \Gamma(s)\, x^{-s}\, ds = e^{-x}.$$
Non-example: the same integrand on the line $\operatorname{Re} s = -\tfrac12$ is not a Barnes integral in this sense, since the pole $s = 0$ lies on the wrong side.
Its value is $e^{-x} - 1$, because the residue $1$ at $s = 0$ is lost.
Barnes' representation of the [[hypergeometric function|463X5V31D9]] reads, for $\abs{\arg(-z)} < \pi$ and $a, b$ not in $\set{0, -1, -2, \dots}$,
$$\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\, {}_2F_1(a, b; c; z) = \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a + s)\Gamma(b + s)\Gamma(-s)}{\Gamma(c + s)} (-z)^s\, ds,$$
with the path separating the poles of $\Gamma(a + s)\Gamma(b + s)$ from those of $\Gamma(-s)$.
The Meijer G-function and the Fox H-function are defined by integrals of this type.